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PreprintCC BY-NC-SA 4.0arXiv

Weighted L2 bounds for kinetic Fokker-Planck equations and hypocoercivity

  • Émeric Bouin
  • Jean Dolbeault
  • Luca Ziviani
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Abstract

We study the long-time behaviour of solutions to kinetic Fokker-Planck equations with power law confinement potentials and local equilibria with fat tails. Without relying on \emph{a priori} moment bounds or perturbative regimes, we establish global estimates on weighted norms (main result) using two methods: a coupling of the weighted norms with entropy dissipation (DMS method) and a Foster-Lyapunov condition approach. As a consequence, we establish an entropy - entropy production inequality and the convergence rates to equilibrium in terms of the exponents associated to the growth of the spatial confinement and the tail decay of local equilibria. For sake of simplicity, we assume that stationary states are factorized.

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Cite

BibTeXRIS
BibTeX
@misc{bouin2026weighted,
  title = {{Weighted $L^2$ bounds for kinetic Fokker-Planck equations and hypocoercivity}},
  author = {Émeric Bouin and Jean Dolbeault and Luca Ziviani},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28120},
  eprint = {2608.28120},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28120},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Weighted L^2 bounds for kinetic Fokker-Planck equations and hypocoercivity
AU  - Émeric Bouin
AU  - Jean Dolbeault
AU  - Luca Ziviani
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28120
UR  - https://arxiv.org/abs/2608.28120
AB  - We study the long-time behaviour of solutions to kinetic Fokker-Planck equations with power law confinement potentials and local equilibria with fat tails. Without relying on \emph{a priori} moment bounds or perturbative regimes, we establish global estimates on weighted norms (main result) using two methods: a coupling of the weighted norms with entropy dissipation (DMS method) and a Foster-Lyapunov condition approach. As a consequence, we establish an entropy - entropy production inequality and the convergence rates to equilibrium in terms of the exponents associated to the growth of the spatial confinement and the tail decay of local equilibria. For sake of simplicity, we assume that stationary states are factorized.
ER  -